Annotation of rpl/lapack/lapack/dsyrfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
! 2: $ X, LDX, FERR, BERR, WORK, IWORK, INFO )
! 3: *
! 4: * -- LAPACK routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
! 10: *
! 11: * .. Scalar Arguments ..
! 12: CHARACTER UPLO
! 13: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IPIV( * ), IWORK( * )
! 17: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
! 18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * DSYRFS improves the computed solution to a system of linear
! 25: * equations when the coefficient matrix is symmetric indefinite, and
! 26: * provides error bounds and backward error estimates for the solution.
! 27: *
! 28: * Arguments
! 29: * =========
! 30: *
! 31: * UPLO (input) CHARACTER*1
! 32: * = 'U': Upper triangle of A is stored;
! 33: * = 'L': Lower triangle of A is stored.
! 34: *
! 35: * N (input) INTEGER
! 36: * The order of the matrix A. N >= 0.
! 37: *
! 38: * NRHS (input) INTEGER
! 39: * The number of right hand sides, i.e., the number of columns
! 40: * of the matrices B and X. NRHS >= 0.
! 41: *
! 42: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
! 43: * The symmetric matrix A. If UPLO = 'U', the leading N-by-N
! 44: * upper triangular part of A contains the upper triangular part
! 45: * of the matrix A, and the strictly lower triangular part of A
! 46: * is not referenced. If UPLO = 'L', the leading N-by-N lower
! 47: * triangular part of A contains the lower triangular part of
! 48: * the matrix A, and the strictly upper triangular part of A is
! 49: * not referenced.
! 50: *
! 51: * LDA (input) INTEGER
! 52: * The leading dimension of the array A. LDA >= max(1,N).
! 53: *
! 54: * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
! 55: * The factored form of the matrix A. AF contains the block
! 56: * diagonal matrix D and the multipliers used to obtain the
! 57: * factor U or L from the factorization A = U*D*U**T or
! 58: * A = L*D*L**T as computed by DSYTRF.
! 59: *
! 60: * LDAF (input) INTEGER
! 61: * The leading dimension of the array AF. LDAF >= max(1,N).
! 62: *
! 63: * IPIV (input) INTEGER array, dimension (N)
! 64: * Details of the interchanges and the block structure of D
! 65: * as determined by DSYTRF.
! 66: *
! 67: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 68: * The right hand side matrix B.
! 69: *
! 70: * LDB (input) INTEGER
! 71: * The leading dimension of the array B. LDB >= max(1,N).
! 72: *
! 73: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 74: * On entry, the solution matrix X, as computed by DSYTRS.
! 75: * On exit, the improved solution matrix X.
! 76: *
! 77: * LDX (input) INTEGER
! 78: * The leading dimension of the array X. LDX >= max(1,N).
! 79: *
! 80: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 81: * The estimated forward error bound for each solution vector
! 82: * X(j) (the j-th column of the solution matrix X).
! 83: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 84: * is an estimated upper bound for the magnitude of the largest
! 85: * element in (X(j) - XTRUE) divided by the magnitude of the
! 86: * largest element in X(j). The estimate is as reliable as
! 87: * the estimate for RCOND, and is almost always a slight
! 88: * overestimate of the true error.
! 89: *
! 90: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 91: * The componentwise relative backward error of each solution
! 92: * vector X(j) (i.e., the smallest relative change in
! 93: * any element of A or B that makes X(j) an exact solution).
! 94: *
! 95: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
! 96: *
! 97: * IWORK (workspace) INTEGER array, dimension (N)
! 98: *
! 99: * INFO (output) INTEGER
! 100: * = 0: successful exit
! 101: * < 0: if INFO = -i, the i-th argument had an illegal value
! 102: *
! 103: * Internal Parameters
! 104: * ===================
! 105: *
! 106: * ITMAX is the maximum number of steps of iterative refinement.
! 107: *
! 108: * =====================================================================
! 109: *
! 110: * .. Parameters ..
! 111: INTEGER ITMAX
! 112: PARAMETER ( ITMAX = 5 )
! 113: DOUBLE PRECISION ZERO
! 114: PARAMETER ( ZERO = 0.0D+0 )
! 115: DOUBLE PRECISION ONE
! 116: PARAMETER ( ONE = 1.0D+0 )
! 117: DOUBLE PRECISION TWO
! 118: PARAMETER ( TWO = 2.0D+0 )
! 119: DOUBLE PRECISION THREE
! 120: PARAMETER ( THREE = 3.0D+0 )
! 121: * ..
! 122: * .. Local Scalars ..
! 123: LOGICAL UPPER
! 124: INTEGER COUNT, I, J, K, KASE, NZ
! 125: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
! 126: * ..
! 127: * .. Local Arrays ..
! 128: INTEGER ISAVE( 3 )
! 129: * ..
! 130: * .. External Subroutines ..
! 131: EXTERNAL DAXPY, DCOPY, DLACN2, DSYMV, DSYTRS, XERBLA
! 132: * ..
! 133: * .. Intrinsic Functions ..
! 134: INTRINSIC ABS, MAX
! 135: * ..
! 136: * .. External Functions ..
! 137: LOGICAL LSAME
! 138: DOUBLE PRECISION DLAMCH
! 139: EXTERNAL LSAME, DLAMCH
! 140: * ..
! 141: * .. Executable Statements ..
! 142: *
! 143: * Test the input parameters.
! 144: *
! 145: INFO = 0
! 146: UPPER = LSAME( UPLO, 'U' )
! 147: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 148: INFO = -1
! 149: ELSE IF( N.LT.0 ) THEN
! 150: INFO = -2
! 151: ELSE IF( NRHS.LT.0 ) THEN
! 152: INFO = -3
! 153: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 154: INFO = -5
! 155: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
! 156: INFO = -7
! 157: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 158: INFO = -10
! 159: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 160: INFO = -12
! 161: END IF
! 162: IF( INFO.NE.0 ) THEN
! 163: CALL XERBLA( 'DSYRFS', -INFO )
! 164: RETURN
! 165: END IF
! 166: *
! 167: * Quick return if possible
! 168: *
! 169: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 170: DO 10 J = 1, NRHS
! 171: FERR( J ) = ZERO
! 172: BERR( J ) = ZERO
! 173: 10 CONTINUE
! 174: RETURN
! 175: END IF
! 176: *
! 177: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 178: *
! 179: NZ = N + 1
! 180: EPS = DLAMCH( 'Epsilon' )
! 181: SAFMIN = DLAMCH( 'Safe minimum' )
! 182: SAFE1 = NZ*SAFMIN
! 183: SAFE2 = SAFE1 / EPS
! 184: *
! 185: * Do for each right hand side
! 186: *
! 187: DO 140 J = 1, NRHS
! 188: *
! 189: COUNT = 1
! 190: LSTRES = THREE
! 191: 20 CONTINUE
! 192: *
! 193: * Loop until stopping criterion is satisfied.
! 194: *
! 195: * Compute residual R = B - A * X
! 196: *
! 197: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
! 198: CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
! 199: $ WORK( N+1 ), 1 )
! 200: *
! 201: * Compute componentwise relative backward error from formula
! 202: *
! 203: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
! 204: *
! 205: * where abs(Z) is the componentwise absolute value of the matrix
! 206: * or vector Z. If the i-th component of the denominator is less
! 207: * than SAFE2, then SAFE1 is added to the i-th components of the
! 208: * numerator and denominator before dividing.
! 209: *
! 210: DO 30 I = 1, N
! 211: WORK( I ) = ABS( B( I, J ) )
! 212: 30 CONTINUE
! 213: *
! 214: * Compute abs(A)*abs(X) + abs(B).
! 215: *
! 216: IF( UPPER ) THEN
! 217: DO 50 K = 1, N
! 218: S = ZERO
! 219: XK = ABS( X( K, J ) )
! 220: DO 40 I = 1, K - 1
! 221: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
! 222: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
! 223: 40 CONTINUE
! 224: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
! 225: 50 CONTINUE
! 226: ELSE
! 227: DO 70 K = 1, N
! 228: S = ZERO
! 229: XK = ABS( X( K, J ) )
! 230: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
! 231: DO 60 I = K + 1, N
! 232: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
! 233: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
! 234: 60 CONTINUE
! 235: WORK( K ) = WORK( K ) + S
! 236: 70 CONTINUE
! 237: END IF
! 238: S = ZERO
! 239: DO 80 I = 1, N
! 240: IF( WORK( I ).GT.SAFE2 ) THEN
! 241: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
! 242: ELSE
! 243: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
! 244: $ ( WORK( I )+SAFE1 ) )
! 245: END IF
! 246: 80 CONTINUE
! 247: BERR( J ) = S
! 248: *
! 249: * Test stopping criterion. Continue iterating if
! 250: * 1) The residual BERR(J) is larger than machine epsilon, and
! 251: * 2) BERR(J) decreased by at least a factor of 2 during the
! 252: * last iteration, and
! 253: * 3) At most ITMAX iterations tried.
! 254: *
! 255: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
! 256: $ COUNT.LE.ITMAX ) THEN
! 257: *
! 258: * Update solution and try again.
! 259: *
! 260: CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
! 261: $ INFO )
! 262: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
! 263: LSTRES = BERR( J )
! 264: COUNT = COUNT + 1
! 265: GO TO 20
! 266: END IF
! 267: *
! 268: * Bound error from formula
! 269: *
! 270: * norm(X - XTRUE) / norm(X) .le. FERR =
! 271: * norm( abs(inv(A))*
! 272: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
! 273: *
! 274: * where
! 275: * norm(Z) is the magnitude of the largest component of Z
! 276: * inv(A) is the inverse of A
! 277: * abs(Z) is the componentwise absolute value of the matrix or
! 278: * vector Z
! 279: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 280: * EPS is machine epsilon
! 281: *
! 282: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
! 283: * is incremented by SAFE1 if the i-th component of
! 284: * abs(A)*abs(X) + abs(B) is less than SAFE2.
! 285: *
! 286: * Use DLACN2 to estimate the infinity-norm of the matrix
! 287: * inv(A) * diag(W),
! 288: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
! 289: *
! 290: DO 90 I = 1, N
! 291: IF( WORK( I ).GT.SAFE2 ) THEN
! 292: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
! 293: ELSE
! 294: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
! 295: END IF
! 296: 90 CONTINUE
! 297: *
! 298: KASE = 0
! 299: 100 CONTINUE
! 300: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
! 301: $ KASE, ISAVE )
! 302: IF( KASE.NE.0 ) THEN
! 303: IF( KASE.EQ.1 ) THEN
! 304: *
! 305: * Multiply by diag(W)*inv(A').
! 306: *
! 307: CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
! 308: $ INFO )
! 309: DO 110 I = 1, N
! 310: WORK( N+I ) = WORK( I )*WORK( N+I )
! 311: 110 CONTINUE
! 312: ELSE IF( KASE.EQ.2 ) THEN
! 313: *
! 314: * Multiply by inv(A)*diag(W).
! 315: *
! 316: DO 120 I = 1, N
! 317: WORK( N+I ) = WORK( I )*WORK( N+I )
! 318: 120 CONTINUE
! 319: CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
! 320: $ INFO )
! 321: END IF
! 322: GO TO 100
! 323: END IF
! 324: *
! 325: * Normalize error.
! 326: *
! 327: LSTRES = ZERO
! 328: DO 130 I = 1, N
! 329: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
! 330: 130 CONTINUE
! 331: IF( LSTRES.NE.ZERO )
! 332: $ FERR( J ) = FERR( J ) / LSTRES
! 333: *
! 334: 140 CONTINUE
! 335: *
! 336: RETURN
! 337: *
! 338: * End of DSYRFS
! 339: *
! 340: END
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